Discrete calculus of variations for quadratic lagrangians

Discrete calculus of v aria tions for quadratic lagrangi ans P . Ryck elynck a, , L. Smo ch a a ULCO, LMP A, F-621 00 Calais, F r anc e Univ Lil le Nor d de F r anc e, F-59000 Lil le, F r anc e. CNRS, FR 2956, F r anc e. Abstract W e develop in this pa p e r a new fra mework for discrete calculus of v ariations when the actions ha ve densities in volving an ar bitrary discretization op er ator. W e deduce the discrete E uler-Lag range equations for piecewise contin uous cr iti- cal p o ints of sampled actions. Then we characterize the discretization o p erators such that, for all quadratic lagrang ian, the discrete Euler-Lag range eq ua tions conv erge to the classical o nes. Keywor ds: Calculus of v aria tions, F unctional equa tio ns, Discretiza tion, Boundary v alue pro blems, Perio dic so lutions 2000 MSC: 49K21 , 49K 15, 65L0 3, 6 5 L12 1. In tro duction Discrete v ar iational problems and disc rete mechanics co nstitute a ctive fields of research a nd ha ve been thor oughly studied by man y authors among which we may cite J. Marsden a nd J. Cresso n, see for instance [1, 2, 5, 6, 7]. A first approach has its ro ots in the w or k o f J . A. Cadzow, developed in the 197 0 s, and consists in repla cing dynamical v ariable s and paths by discr e te v ar iables and paths which are finite linea r combinations of indicator functions. Bas ed on this discretization, the action in tegra l, the lagrangian a nd the energy ar e approxi- mated on a time s lice [ k ε, ( k + 1) ε ], for some time delay ε , by discrete analogs which ar e finite sums dep ending of a finite n umber o f v ariables in R d . The presentation of this theory , together with an history and la r ge bibliography , is developed in [6]. Another different a ppr oach consists in re placing the deriv ative ˙ x ( t ) o f the dynamical v ariable x ( t ) with a three terms scale deriv ative  ε, Q x ( t ). In this wa y , a dynamical v ariable x ( t ) r emains even a fter the sampling pro cess a function  ε, Q x ( t ) o f “ contin uous” time. The principle o f le ast action ma y be extended to the case of no n-differentiable dynamical v ar iables of H¨ olderian regular ity C 1 / 2 . W e refer to [1, 2] for further details among which the ex tension of No ether ’s Theor em and also an in teresting informal discussio n ab out fractal Email addr esses: ryckelyn@l mpa.univ-littoral.fr (P . Ryck elync k), smoch@lm pa.univ-litto ral.fr (L. Smo c h) Pr eprint submitted to Elsevier Novemb e r 23, 2021 ph ysics . While the first appro a ch is widely used in numerical a nalysis of v ar ia- tional dynamical systems, the second one is, roughly sp eak ing, more co ncerned with foundatio nal discussio n of micr ophysics. In this pap er we highlight and use the equations of motion ∇ x L +  − ε ∇ ˙ x L = 0 , (1) for all lagrangian L and genera l discretization op er ator  ε defined a s  ε x ( t ) = N X ℓ = − N c ℓ x ( t + ℓε ) χ − ℓ ( t ) . (2) In this formula, the sy m b ols χ − ℓ ( t ) denote ch ar acteristic functions o f some int erv als, a nd preven t t + ℓε from belong ing to a n in terv al in whic h x is undefined. A first motiv ation in this pap er is to justify the c hoice of  ε, Q in [1, 2]. T o do this, w e use throug hout the previous generalizatio n (2) and distinguish among the op era tors  ε those having s ome sp ecific features. A seco nd intend is to understa nd the similar ities and differences betw een the equa tions o f motions for disc retized dyna mical v ariables and the classical Euler-La grang e eq uations L ′ x − d/dt L ′ ˙ x = 0. A third one is to generalize some results of [1, 2] to op er a tors  ε not satisfying so me Leibniz formula and to extremal curves x ( t ) which are not C 1 / 2 but ra ther of a weaker sp ecific regular ity C pw . In Sectio n 2, we first fix the notation use d in the pap er and study tw o sp ecific sets O N ,ε and ˜ O N ,ε of op erator s (2). In Section 3, we descr ibe a class of op erator s  ε satisfying an appropr ia te extension o f Leibniz fo rmula. In Section 4, we get the necessary first o rder co ndition for finding a minimizer o f a discr ete action A disc ( x ). W e prove the equa tio n (1), whic h dep ends only on  ε and is reminiscent of Cr e s son’s res ult. In Section 5, we first in tro duce the classic a l and discrete mo dels for the quadra tic lagr angians and emphasize on their similarity . Next, the subset of op erato rs  ε for which discrete Euler Lagr ange equa tions hav e oscillator y solutions is examined. In Section 6, we state the ma in r esult of the pap e r which characterizes the set o f o p e rators  ε for which the conv erg ence of (1) holds, for e very q uadratic lagr angian. In this paper, C.E .L./D.E.L. stand for classical/discre te Euler-La grange equations. 2. Notation and framew ork First, let us fix some no tation. L e t [ a, b ] b e so me interv al of time and a time delay ε > 0 be fixe d thro ughout. W e deno te by d the “physical” dimension and by N the num be r of samples in C d . W e us e the nota tion i for √ − 1. W e deno te by χ ℓ ( t ) the characteristic function o f the interv al [max( a, a + ℓε ) , min( b, b + ℓε )], for all in teger ℓ . Many int ere sting ways to generate discrete actions aris e in the fo llowing w ay . The der iv ative ˙ x ( t ) o ccuring in la g rangia ns is replaced with a linear c o mbination 2 of discretized v alues of x ( t )  [ r,s ] ε x ( t ) = − χ +1 ( t ) s ε x ( t − ε ) + s − r ε x ( t ) + χ − 1 ( t ) r ε x ( t + ε ) (3) where r, s ∈ C ⋆ . In [1, 2], the a uthors c ho os e the v alues r = (1 − i ) / 2 and s = (1 + i ) / 2 but do not work with characteristic functions. Let  ε, Q be defined up to no w by  [ 1 − i 2 1+ i 2 ] ε . Obviously , (3) is a particular case of (2 ) with N = 1 , s = − c − 1 ε , r = c 1 ε and c 0 = − ( c − 1 + c 1 ). W e generalize this process by using linear combinations o f 2 N + 1 terms o f the s hap e c ℓ x ( t + ℓ ε ) χ − ℓ ( t ) where c ℓ ∈ C are fixed. Definition 2.1 . Given α and β in C d , the affine sp ac e C pw ( d, N , α, β ) is the set of the functions x : [ a, b ] → C d satisfying x ( a ) = α , x ( b ) = β and c ontinu ou s on e ach interval [ a + ℓε, a + ( ℓ + 1) ε ] ∩ [ a, b ] for al l inte ger ℓ . This space is endowed with the top olo gy of uniform conv erge nc e , its tangent space is ev erywher e the Ba nach space C pw ( d, N , 0 , 0). Definition 2 .2 . F or any x ∈ C pw ( d, N , α, β ) , we denote by S ( x ) the r ow ve ctor value d function S ( x )( t ) = ( x 1 ( t − N ε ) χ N ( t ) . . . x j ( t + ℓε ) χ − ℓ ( t ) . . . x d ( t + N ε ) χ − N ( t )) . (4) In S ( x ) the ordering of v ar ia bles is le xicogra phic, first o n ℓ and next on j . The function S ( x ) lies in a pro duct of d (2 N + 1) affine spaces modeled on C pw (1 , N , 0 , 0) and, in this w ay , S is an injective contin uous linear mapping. Definition 2.3 . If ( c ℓ ) ∈ C 2 N +1 , t he gener alize d sc ale derivative is the c ontinuous line ar endomorphism of C pw ( d, N , α, β ) define d by (2) for al l t ∈ [ a, b ] . If d = 1 , for all x ∈ C pw (1 , N , α, β ), w e have  ε x = ( c − N . . . c 0 . . . c N ) t ( S ( x )). When d ≥ 2, we hav e a simila r relationship b etw een  ε x and S ( x ), inv olving a d × d (2 N + 1) banded matrix  ε x =      c − N . . . c 0 . . . c N c − N . . . c 0 . . . c N 0 0 . . . c − N . . . c 0 . . . c N      t ( S ( x )) . Note that  ε is a w ell defined contin uous e ndo morphism of the Ba nach space C pw ( d, N , α, β ). The conv ergence of  ε x to the ordinary deriv ative ˙ x is connected to th e following definitions. Definition 2.4 . The ve ctor sp ac e O N ,ε is the set of op er ators  ε of the shap e (2) with c o efficients c ℓ = γ ℓ /ε and whe r e γ ℓ ∈ C do not dep end on ε . Algebr aic al ly, O N ,ε ≃ C 2 N +1 . 3 Definition 2 .5 . The affine sp ac e ˜ O N ,ε ⊂ O N ,ε c ontains op er ators  ε for which  ε 1 = 0 and  ε t = 1 when t lies in the safety interval I S = [ a + 2 N ε, b − 2 N ε ] . W e shall see , as a consequence of T he o rem 6.1, that if  ε ∈ ˜ O N ,ε , then  ε x ( t ) tends to ˙ x ( t ) lo cally uniformly in ] a, b [, for all x ∈ C 2 ([ a, b ] , C d ). R emark 2.1 . The discrete Euler forward a nd ba ckw ard difference opera tors are resp ectively defined by ∆ + ε =  [1 , 0] ε and ∆ − ε =  [0 , 1] ε . Their mean is the sym- metric difference opera to r  ε, S =  [ 1 2 , 1 2 ] ε . These three op er ators and  ε, Q are elements of ˜ O 1 ,ε . Notice that the op erator  ε, Q is also related to the one- dimensional v ers ion of the op er a tor used b y Kime [4] when she solv es n umerically the Schro edinger equa tion via F orward-Difference and Backw ard-Differe nce ap- proximations a t v a rious steps in time. R emark 2.2 . The op erator x → ∆ + ε (∆ + ε x ( t − ε )) = 1 ε  [1 , − 1] ε x ( t ) approximates well the second deriv ative but does not lie in O 1 ,ε . 3. Leibniz formulas for  ε op erators in O 1 ,ε In order to deduce his version of D.E.L. with  ε, Q , Cresson in [1] found an analog o f the classica l L e ibniz for mula. When such a formula exists, a pr inciple of discre te virtual works may b e sta ted. In this section, we generalize Cr esson’s ident ity to the family o f op erato rs (3). The or em 3 .1 . L et  ε ∈ O 1 ,ε of the shap e ( 3 ) with r , s ∈ C ⋆ and s/ r / ∈ R then, for al l pie c ewise c ontinuous functions f , g : [ a, b ] → C d , we get the gener alize d L eibniz formula  [ r,s ] ε ( f · g )( t ) = f ( t ) ·  [ r,s ] ε g ( t ) + g ( t ) ·  [ r,s ] ε f ( t )+ ε ( r s 2 − r 2 s ) ( rs − r s ) 2  [ r,s ] ε f ( t ) ·  [ r,s ] ε g ( t ) − εrs ( r − s ) ( rs − r s ) 2  [ r,s ] ε f ( t ) ·  [ r,s ] ε g ( t )+ εrs ( r − s ) ( rs − r s ) 2   [ r,s ] ε f ( t ) ·  [ r,s ] ε g ( t ) +  [ r,s ] ε f ( t ) ·  [ r,s ] ε g ( t )  . (5) Pr o of. The formula (5) is C -bilinear w.r.t. f a nd g . Having in mind the prop- erties of the inner pro duct and the fa c t that  ε acts comp onent-wise, w e may suppo se without loss of generality that d = 1 and f , g ∈ R . W e s lightly g en- eralize the pro of of Theorem 2.1 of [1 ] w hich attempts to give a fo r mula such as W ( f g ) = W ( f ) g + f W ( g )+ 4 d 1 W ( f ) W ( g ) + d 2 ˜ W ( f ) W ( g ) + d 3 W ( f ) ˜ W ( g ) + d 4 ˜ W ( f ) ˜ W ( g ) . (6) Here W and ˜ W are tw o o pe rators in ˜ O 1 ,ε , f and g are arbitra ry in C pw (1 , 1 , α, β ) and d 1 , d 2 , d 3 , d 4 are four complex num be r s. Now, we cho ose W =  [ r,s ] ε and ˜ W =  [ r ′ ,s ′ ] ε for s o me complex n um b ers r , s, r ′ , s ′ such that r s ′ − s r ′ 6 = 0. W e hav e obviously W = r ∆ + ε + s ∆ − ε and ˜ W = r ′ ∆ + ε + s ′ ∆ − ε . But we have also the well-kno wn formulas for ∆ + ε and ∆ − ε ∆ σ ε ( f g ) = ∆ σ ε ( f ) g + f ∆ σ ε ( g ) + σε ∆ σ ε ( f )∆ σ ε ( g ) , (7) where σ = ± 1. Substituting the four previous formulas in (6), the identit y (6) holds if and only if the coefficients d 1 , d 2 , d 3 and d 4 satisfy     r 2 rr ′ rr ′ r ′ 2 rs sr ′ rs ′ r ′ s ′ rs rs ′ sr ′ r ′ s ′ s 2 ss ′ ss ′ s ′ 2         d 1 d 2 d 3 d 4     =     rε 0 0 − sε     (8) whose determinant is equal to − ( rs ′ − sr ′ ) 4 6 = 0. W e replace in (6) the co efficie nts d ℓ by their explicit v alue s d 1 = ε δ ( rs ′ 2 − sr ′ 2 ) , d 2 = d 3 = εrs δ ( r ′ − s ′ ) , d 4 = εrs δ ( s − r ) (9) where δ = ( rs ′ − sr ′ ) 2 . Since s/r / ∈ R , we can c ho ose r ′ = r and s ′ = s and we get eas ily the formula (5). As a n e xample, we g et − d 1 = d 2 = d 3 = d 4 = − 1 2 iε for the o per ator  ε, Q =  [ 1 − i 2 , 1+ i 2 ] ε chosen in [1, 2], so that fo r all piecewise contin uous f , g : [ a, b ] → R  ε, Q ( f g ) =  ε, Q ( f ) g + f  ε, Q ( g ) + 1 2 iε [  ε, Q ( f )  ε, Q ( g ) −  ε, Q ( f ) ⊟ ε, Q ( g ) − ⊟ ε, Q ( f )  ε, Q ( g ) − ⊟ ε, Q ( f ) ⊟ ε, Q ( g )], where ⊟ ε stands for the complex conjugate op erator of  ε . (W e have corr ected here the corresp onding formula in [1, 2 ].) 4. Critical p oin ts of dis crete actions According to the previo us notation, the discrete actions a nd lagr a ngians a re related to classical ones as follows. Definition 4.1 . If L ( t, x ( t ) , ˙ x ( t )) is a lagr angian dep ending on 2 d + 1 vari ables then we set L ( t, S ( x )( t )) = L ( t, x ( t ) ,  ε x ( t )) . Mor e over, A cont ( x ) = Z b a L ( t, x ( t ) , ˙ x ( t )) dt (10) and A disc ( x ) = Z b a L ( t, S ( x )( t )) dt = Z b a L ( t, x ( t ) ,  ε x ( t )) dt (11) ar e the r esp e ctive classic al and discr ete actio ns asso ciate d to these lagr angians, define d r esp e ctively on C 1 ([ a, b ] , C d ) and C pw ( d, N , α, β ) . 5 If the ter ms x j ( t + ℓε ) χ − ℓ ( t ) o ccuring in S ( x )( t ) (s e e (4)) ar e replaced with new v a r iables ξ j,ℓ , then L ( t, ξ 1 , − N , . . . , ξ d,N ) is nothing but L ( t, x ( t ) ,  ε x ( t )) and thus dep ends on d (2 N + 1) + 1 indeterminates. A fundamental problem is to minimize the action (10) under Dirichlet bound- ary conditions when obviously , ev ery unknown and parameter has to b e r eal. Note that in o rder to deal with optima instead of critical p oints of the action (11), we hav e to handle real v alued functions and parameters. The or em 4.1 . L et x ∈ C pw ( d, N , α, β ) b e a critic al p oint of (11). Then x satisfies the fol lowing fun ctional e quation ∀ j ∈ { 1 , . . . , d } , N X ℓ = − N ∂ L ∂ ξ j,ℓ ( t − ℓε, S ( x )( t − ℓ ǫ )) χ ℓ ( t ) = 0 (12) and the e quation (12) may b e r eturne d u nder an intrinsic form as (1), i.e.  − ε ∂ L ∂ ˙ x j ( t, x ( t ) ,  ε x ( t )) + ∂ L ∂ x j ( t, x ( t ) ,  ε x ( t )) = 0 , (13) for al l j ∈ { 1 , . . . , d } . Pr o of. W e hav e for all h ∈ C pw ( d, N , 0 , 0) A disc ( x + η h ) − A disc ( x ) = Z b a ( L ( t, S ( x + η h )( t )) − L ( t, S ( x )( t ))) dt = η Z b a d X j =1 N X ℓ = − N ∂ L ∂ ξ j,ℓ ( t, S ( x )( t )) h j ( t + ℓε ) χ − ℓ ( t ) dt + O ( η 2 ). Therefore using Chasles relation and setting t = τ − ℓε in the previous equality , we find the Gˆ ateaux der iv ative D A disc ( x )( h ) = d X j =1 N X ℓ = − N Z b + ℓε a + ℓε ∂ L ∂ ξ j,ℓ ( τ − ℓ ε, S ( x )( τ − ℓ ε )) h j ( τ ) dτ = Z b + N ε a − N ε d X j =1 h j ( τ ) N X ℓ = − N ∂ L ∂ ξ j,ℓ ( τ − ℓ ε, S ( x )( τ − ℓ ε )) χ ℓ ( τ ) dτ = 0 , which gives (12) by using Paul Dubo is-Reymond lemma in the multidimensional case. By definition of L ( t, S ( x )( t )), w e have for all ℓ 6 = 0, ∂ L ∂ ξ j,ℓ ( t, S ( x )( t )) = ∂ L ∂ ˙ x j ( t, x ( t ) ,  ε x ( t )) c ℓ χ − ℓ and for ℓ = 0, ∂ L ∂ ξ j, 0 ( t, S ( x )( t )) = ∂ L ∂ ˙ x j ( t, x ( t ) ,  ε x ( t )) c 0 + ∂ L ∂ x j ( t, x ( t ) ,  ε x ( t )). Thu s, (12) is equiv alent to 6 N X ℓ = − N c ℓ χ − ℓ ∂ L ∂ ˙ x j ( t − ℓε, x ( t − ℓε ) ,  ε x ( t − ℓε )) + ∂ L ∂ x j ( t, x ( t ) ,  ε x ( t )) = 0. Since the first term is eq ual to  − ε ∂ L ∂ ˙ x j ( t, x ( t ) ,  ε x ( t )), the res ult is proved. R emark 4.1 . When  ε is not o f the shap e (3), we do not hav e (5) and integration by parts ma y not be per formed. W e use instead s imple changes of v aria bles. Note also that seco nd o rder deriv a tives of the lagra ngian o ccuring in C.E .L. are replaced with time delayed first or der o nes. R emark 4.2 . In [1 ], Cresson de a ls with the cas e N = 1 and the op erato r  ε, Q =  [ 1 − i 2 , 1+ i 2 ] ε . F or an y function f ( x , ε ) he defines the ε -dominant part [ f ] ε with a limiting pr o cess. He proved that if lim ε → 0 D A disc ( x ) = 0, then  ∂ L ∂ x j −  ε,q ∂ L ∂ ˙ x j  ε = lim ε → 0  ∂ L ∂ x j −  ε,q ∂ L ∂ ˙ x j  = 0 (14) which is an alter native form of (1 3). Our equation (1 3) is how ever differen t from (14) at le a st in three res pe c ts . First, the characteristic functions χ − ℓ ( t ) of the v arious interv als app ear in the sampling pro cess (2) as w ell as in the ac tio n (11). The second one is that (13) does not dep end o n the co efficients nor o n the length of for mula (2). The last one is the use in (13) of  − ε instead of −  ε . B ut we hav e [  ε,q f ( t )] ε = [ −  − ε,q f ( t )] ε for all r eal v alued function f and t ∈ [ a + ε, b − ε ]. Indeed, [ ℑ (  ε, Q f )] ε = 0 and ℜ (  − ε,q f ) = −ℜ (  ε, Q f ) since every characteristic function in  ε, Q is equal to 1. 5. Quadratic lagrangians in discrete and classical s ettings In this s e ction we dea l w ith a system o f d or dinary differential equa tions of the sec ond orde r arising from the following lag rangia n L ( t, x , ˙ x ) = 1 2 t ˙ x P ˙ x + 1 2 t x Q x + t x R ˙ x + t J 1 ˙ x + t J 2 x + J 3 , (15 ) where P ( t ) , Q ( t ) , R ( t ) ∈ C d × d , J 1 ( t ) , J 2 ( t ) ∈ C d and J 3 ( t ) is a sc alar function. Many physical s y stems might b e mo delized by such lagra ngians, in electro ma g- netism, qua nt um mechanics, mater ial science, regulators mo dels and so o n. A co nv enient setup that we assume from now on is that the coefficients in (15) are rea l and smo oth, a nd that for all t ∈ [ a, b ], P ( t ) and Q ( t ) a re symmetric and R ( t ) is skew-symmetric (the sy mmetric part of R ( t ) gives rise to a null lagrang ian term in (15)). 7 The or em 5.1 . L et L b e a quadr atic lagr angian such t hat t P = P , t Q = Q , t R = − R and L asso ciate d t o L as in (11). The Euler-L agr ange e quation asso ciate d to (15) c an b e written as − P ¨ x + ( − ˙ P + 2 R ) ˙ x + ( ˙ R + Q ) x − ˙ J 1 + J 2 = 0 . (16) The e quation (16) may b e discr etize d a p osteriori to give − P  ε (  ε x ) + ( − ˙ P + 2 R )  ε x + ( ˙ R + Q ) x − ˙ J 1 + J 2 = 0 . (17) If x ∈ C pw ( d, N , α, β ) is a critic al p oint of the action (11), t hen it must satisfy  − ε ( P  ε x ) −  − ε ( R x ) + R  ε x + Q x +  − ε J 1 + J 2 = 0 . (18) Pr o of. First, (1 6) is straightforw ard, since we get ∂ L ( t, x , ˙ x ) ∂ x = Q x + R ˙ x + J 2 and ∂ L ( t, x , ˙ x ) ∂ ˙ x = P ˙ x − R x + J 1 . Next, (17) is obtained by discr etizing the deriv atives in (16), i.e. by re pla c- ing ˙ x , ¨ x by  ε x ,  ε (  ε x ) respectively and the result holds. At las t, (18) is a consequence of (13). Indeed, using the pr evious deriv atives, (13) gives  − ε ( P  ε x − R x + J 1 ) + Q x + R  ε x + J 2 = 0 , which ends the pro o f. R emark 5.1 . If P , Q, R ar e indep endent on time then, for t ∈ I S , b oth equations (18) and (17) ar e eq uiv alent. R emark 5.2 . F or the so-called linear quadratic control problem, Guib out a nd Scheeres [3] give Euler-La grang e different ial equations having a similar structure than (16). They suppose that the relation ˙ x ( t ) = M 1 ( t ) x ( t ) + M 2 ( t ) u ( t ) holds betw een the dynamical v ar ia ble x and the control v ar iable u and prove that each cr itical p oint ( x , p ) satisfies ˙ x ( t ) = ( M 1 − 1 2 M 2 P − 1 t R ) x ( t ) − M 2 P − 1 t M 2 p ( t ), ˙ p ( t ) = ( 1 2 R t P − 1 t M 2 − t M 1 ) p ( t ) + ( 1 4 RP − 1 t R − Q ) x ( t ), where p stands for a Lagra nge multiplier of the constra int. Let us inv estigate no w the ha r monic oscillator. W e ha ve in this case d = 1, R = 0, J 1 = J 2 = 0 and we set moreov er P ( t ) = p , Q ( t ) = q with pq < 0. Solutions of C.E .L. − p ¨ x + q x = 0 ar e perio dic. Let us s tudy the p erio dicit y of solutions of D.E .L . for ε small enough. If t lies in I S , (21) ma y be simplified int o pc − 1 c 1 [ x ( t − 2 ε ) + x ( t + 2 ε )] + p c 0 ( c − 1 + c 1 )[ x ( t − ε ) + x ( t + ε )] + ( q + p ( c 2 − 1 + c 2 0 + c 2 1 )) x ( t ) = 0 . (19) The result below presents an a dditional characteriza tion of “suitable”  ε . 8 Pr op osition 5.1 . L et  ε ∈ ˜ O 1 ,ε . The two fol lowing pr op erties ar e e quivalent. (a) F or al l p and q with pq < 0 and for al l ε smal l enou gh, the r o ots of t he char acteristic p olynomial of (19) ar e of mo du lu s 1. (b) F or some k ∈ R , we have  ε =  [ 1 2 , 1 2 ] ε + ik  [1 , − 1] ε . Pr o of. The c hara cteristic p olynomia l of the recurrence (19) is symmetric that is to say D ( λ ) = λ 4 D (1 /λ ). Setting µ = λ + 1 /λ , we get a quadra tic E ( µ ) such that E ( µ ) = ε 2 D ( λ ) = γ 1 γ − 1 µ 2 + γ 0 ( γ 1 + γ − 1 ) µ + ( γ 2 0 + γ 2 1 + γ 2 − 1 − 2 γ 1 γ − 1 + q p ε 2 ) . ( a ) ⇒ ( b ). W e loo k for the pa rameters γ − 1 , γ 0 , γ 1 for whic h D ( λ ) has ro o ts on the unit circle, for ε small enough and for all p, q , p q < 0. This amounts to sa y that E ( µ ) has only real ro o ts in [ − 2 , 2] for all p, q , pq < 0, provided ε is small enough. Let us gener alize a little bit. Let α, β , γ be three complex num b er s, then the solutions of the quadratic αy 2 + β y + γ + ε = 0 a r e in [ − 2 , 2] ⊂ R for all ε small enough if and o nly if we hav e α, β , γ ∈ R and | β | ≤ 4 | α | , | γ | ≤ 4 | α | , 4 αγ ≤ β 2 , 8 | β | ≤ 1 6 | α | + 4 γ sg n ( α ) , as sho ws explicit computations us ing the usual s olution of a quadr atic. Since  ε ∈ ˜ O 1 ,ε , w e ha ve  ε 1 = γ − 1 + γ 0 + γ 1 = 0 and  ε t = γ 1 − γ − 1 = 1 inside I S if and only if γ 1 = r , γ − 1 = r − 1 and γ 0 = − 2 r + 1 for so me r ∈ C . So, α = r ( r − 1), β = (2 r − 1) 2 and γ = 4 r 2 − 4 r + 2 are rea l. Ther efore we o btain ℜ ( r ) = 1 / 2 a nd easy computations lead to ( b ). ( b ) ⇒ ( a ). If ( b ) holds then setting r = 1 2 + ik , w e get γ 1 = r , γ − 1 = r − 1 and γ 0 = − 2 r + 1. Direct co mputations show that the solutions of E ( µ ) = r ( r − 1) µ 2 − (2 r − 1) 2 µ + (4 r 2 − 4 r + 2) − ω 2 ε 2 , (20) where ω 2 = − q / p , ar e in [ − 2 , 2] ⊂ R for all ε ≤ 1 / ( | ω | √ 1 + 4 k 2 ). R emark 5.3 . W e r ecov er  ε, Q and  ε, S by s etting k = − 1 / 2 and k = 0 in ( b ) resp ectively . R emark 5.4 . If  ε lies in O 1 ,ε , the prop erty ( a ) is equiv a lent to the following inequalities for γ − 1 , γ 0 , γ 1 ∈ C 4 | γ 1 γ − 1 | − | γ 0 ( γ 1 + γ − 1 ) | ≥ 0, 4 | γ 1 γ − 1 | − | γ 2 0 + γ 2 1 + γ 2 − 1 − 2 γ 1 γ − 1 | ≥ 0, γ 2 0 ( γ 1 + γ − 1 ) 2 − 4 γ 1 γ − 1 ( γ 2 0 + γ 2 1 + γ 2 − 1 − 2 γ 1 γ − 1 ) ≥ 0 a nd 16 | γ 1 γ − 1 | + 4 γ 2 0 + 4 γ 2 1 + 4 γ 2 − 1 − 8 γ 1 γ − 1 − 8 | γ 0 ( γ 1 + γ − 1 ) | ≥ 0. So there exists op er ators  ε / ∈ ˜ O 1 ,ε satisfying ( a ) but not ( b ). 9 6. Con v ergence of functional equations D. E.L. to diffe rential equa- tions C.E.L. In this section, w e addre ss the problem o f co nv e rgence of D.E.L. to C.E.L. in the case of quadratic la grangia ns. Roughly , this con vergence pro p erty char- acterizes ˜ O N ,ε . In the remainder, the matrices P , Q, R in (15) are dep endent on time, P, Q are symmetric and R is skew-symmetric. T o b egin with, let us give the following L emma 6.1 . The left hand side of D.E.L. (18) is e qual to Θ( x )( t ) = X − 2 N ≤ ℓ ≤ 2 N − N ≤ j ≤ N | ℓ + j | ≤ N c ℓ + j c j χ j ( t ) χ − ℓ ( t ) P ( t − j ε ) x ( t + ℓε ) + Q ( t ) x ( t )+ N X ℓ = − N χ − ℓ ( t )( c ℓ R ( t ) − c − ℓ R ( t + ℓε )) x ( t + ℓε ) +  − ε J 1 ( t ) + J 2 ( t ) (21) for al l t ∈ [ a, b ] . Pr o of. The pr o of is straightforw ard by using several times the formula (2) for  ε and  − ε , and showing that (18) may b e retur ned as Θ( x )( t ) = 0 fo r all t ∈ [ a, b ]. Definition 6.1 . We say that D.E.L. ( 18) c onver ges to C.E.L. (16) as ε tends to 0 if, for al l quadr atic lagr angian L as in (15) and al l x ∈ C 2 ([ a, b ] , R d ) , lim ε → 0 Θ( x )( t ) = − P ¨ x + ( − ˙ P + 2 R ) ˙ x + ( ˙ R + Q ) x − ˙ J 1 + J 2 lo c al ly uniformly in ] a, b [ , in the norm of L ∞ ([ a + δ , b − δ ]) for all δ > 0 . The remainder o f this section is devoted to the pro o f of the main r esult below. The or em 6.1 . Le t  ε ∈ O N ,ε . The fol lowing six pr op erties ar e e quivalent. (a) D.E.L. (18) c onver ges to C.E.L. (16) as ε tends to 0. (b) F or al l x ∈ C 2 ([ a, b ] , R d ) , lim ε → 0  ε x ( t ) = ˙ x ( t ) lo c al ly uniformly in ] a , b [ . (c) F or al l x ∈ C 2 ([ a, b ] , R d ) , lim ε → 0  − ε x ( t ) = − ˙ x ( t ) lo c al ly un iformly in ] a, b [ . (d) The functions t →  ε 1 and t →  ε t c onver ge r esp e ctively to 0 and 1 lo c al ly uniformly in ] a, b [ . (e)  ε ∈ ˜ O N ,ε . (f ) Ther e exists c omplex n u mb ers k 1 , . . . , k 2 N − 1 such that  ε x ( t ) =  [1 , 0] ε x ( t ) + N − 1 X ℓ = − ( N − 1) k ℓ + N  [1 , − 1] ε x ( t − ℓε ) . (22) 10 Pr o of. W e will prov e that ( a ) ⇒ ( c ) ⇒ ( d ) ⇔ ( e ) ⇔ ( f ), ( d ) ⇒ ( b ) ⇒ ( c ) ⇒ ( a ). ( a ) ⇒ ( c ). By assumption, the function Θ(0) =  − ε J 1 + J 2 m ust tend to the constant ter m in (16) that is − ˙ J 1 + J 2 , for all J 1 and J 2 , a nd the result holds. ( c ) ⇒ ( d ). W e cho ose x a s a linea r function t → x 1 t + x 0 which c hecks the prop erty in ( c ). As  − ε acts co mpo nent-wise,  − ε 1 a nd  − ε t tend to 0 as ε tends to 0. But since  ε 1 = 1 ε N X ℓ = − N γ ℓ χ − ℓ ( t ) and  ε t = t  ε 1 + 1 2 N X ℓ = − N ℓ ( γ ℓ χ − ℓ ( t ) − γ − ℓ χ ℓ ( t )) , (23) for all ε 6 = 0, we hav e inside I S the relatio ns  ε 1 = −  − ε 1 and  ε t = −  − ε t + t (  ε 1 +  − ε 1). This s hows that ( d ) holds. ( d ) ⇔ ( e ). F o r all δ > 0 a nd ε s mall enoug h, if t ∈ [ a + δ, b − δ ] then t ∈ I S , and we see that the as sumptions in ( d ) a re equiv a lent to (23) and in tur n to the tw o linear eq uations N X ℓ = − N γ ℓ = 0 and 1 2 N X ℓ = − N ℓ ( γ ℓ − γ − ℓ ) = 1 . (24) If  ε ∈ O N ,ε satisfies these tw o equations then  ε 1 → 0 and  ε t → 1 unifor mly lo cally in ] a, b [, tha t is  ε satisfies the s tatement s in ( d ) and c onv ersely . Thus, we hav e proved that  ε ∈ ˜ O N ,ε if and only if ( d ) holds. ( e ) ⇔ ( f ). The t wo equations (24) b eing indep endent, co dim( ˜ O N ,ε ) = 2 . If ˆ O N ,ε is the s et of o p erators defined by (2 2), then c o dim( ˆ O N ,ε ) = 2 . W e easily see that each op erator  ε ∈ ˆ O N ,ε chec k s the tw o equations in ( d ), that is ˆ O N ,ε ⊂ ˜ O N ,ε and lastly , ˆ O N ,ε = ˜ O N ,ε . ( d ) ⇒ ( b ). W e use T aylor-Lagr ange formula to obtain an expansio n of  ε x as  ε x ( t ) = N X ℓ = − N c ℓ χ − ℓ ( t ) " x ( t ) + ℓε ˙ x ( t ) + Z t + ℓε t ( t + ℓε − s ) ¨ x ( s ) ds # = (  ε 1) x ( t ) + (  ε t − t  ε 1) ˙ x ( t ) + Z b a G ( s, t ) ¨ x ( s ) ds. By using ˜ χ [0 ,ℓε ] = χ [0 ,ℓε ] if ℓ ≥ 0 and ˜ χ [0 ,ℓε ] = χ [ ℓε, 0] if ℓ ≤ 0, the previo us kernel G ( s, t ) is equal to G ( s, t ) = χ 0 ( s ) χ 0 ( t ) N X ℓ = − N ( t + ℓε − s ) c ℓ χ − ℓ ( t ) ˜ χ [0 ,ℓε ] ( s − t ) . (25) Let δ > 0 , V 1 = L ∞ ([ a, b ]), V 2 = L ∞ ([ a + δ, b − δ ]) and V 3 = L ∞ ([ a, b ] × [ a + δ, b − δ ]). Then, fo r all function x ∈ C 2 ([ a, b ]), if ε < δ / N , the no rm k  ε x − ˙ x k V 2 is b ounded b y k  ε 1 k V 2 ( k x k V 1 + max( | a | , | b | ) k ˙ x k V 1 ) + k  ε t − 1 k V 2 k ˙ x k V 1 + ( b − a ) k G k V 3 k ¨ x k V 1 . 11 The first and the second terms conv erg e to 0 as ε tends to 0 help to ( d ). In order to pro ve that the last term tends also to 0 a s ε tends to 0, we note that G ( s, t ) = ( t − s )  ε 1 + N X ℓ = − N ( ℓε ˜ χ [0 ,ℓε ] ( s − t ) + ( t − s )( ˜ χ [0 ,ℓε ] ( s − t ) − 1 )) c ℓ χ − ℓ ( t ) . Obviously , first and second terms tend to 0 as ε tends to 0. T o deal with the third term, we dis tinguish tw o cas es. If t ≤ s ≤ t + k ε for some k ∈ [ − N , N ], the term is b ounded by N ε P N ℓ = − N | c ℓ | while if | s − t | > N ε , it is b ounded by | s − t | . ||  ε 1 || V 2 , and th us, by the sum of these t wo upp er b ounds. Collecting altogether, we find k G k V 3 ≤ 2 max( | b − a | , 2 | a | , 2 | b | ) k  ε 1 k V 2 + N ε N X ℓ = − N | c ℓ | + ε N X ℓ = − N | ℓc ℓ | . Therefore, k G k V 3 tends to 0 and thus, also k  ε x − ˙ x k V 1 as ε tends to 0 and the result holds. ( b ) ⇒ ( c ). The tw o op era tors  ε and  − ε are linked b y the fo llowing prop erty . If x 1 ∈ C pw ( d, N , α, β ) then the function x 2 defined b y x 2 ( t ) = x 1 ( a + b − t ) is in C pw ( d, N , β , α ) and satisfies (  − ε x 1 )( t ) = − (  ε x 2 )( a + b − t ) . Indeed, this formula comes from (2) and the fact that χ − j ( s ) = χ j ( a + b − s ) for all j . Since ˙ x 2 ( t ) = − ˙ x 1 ( a + b − t ), the conv erge nce o f  ε x 2 to ˙ x 2 for all x 2 ∈ C pw ( d, N , β , α ) implies the conv erg e nc e of  − ε x 1 to − ˙ x 1 for all x 1 ∈ C pw ( d, N , α, β ). ( c ) ⇒ ( a ). W e use ag ain the form ula (21) and we deal with its non-constant terms. When we develop x ( t + ℓε ) help to the T a ylor -Mac Laur in form ula of order 2, w e get Θ( x )( t ) = − P ε ( t ) ¨ x ( t )+ 2 R ε ( t ) ˙ x ( t )+ Q ε ( t ) x ( t )+  − ε J 1 ( t )+ J 2 ( t )+ ˜ Θ( x )( t ) , (26) where nota tion is explained hereafter. First, let us note that ˜ Θ( x )( t ) is a com- bination of tw o kinds of terms M ( ¨ x ( t + θ ℓ ℓε ) − ¨ x ( t )), for some θ ℓ ∈ ]0 , 1[ and ℓ ∈ {− 2 N , . . . , 2 N } , M being e ither e qual to P ( t − j ε ) or R ( t + ℓε ). When t ∈ I S , we see that those terms tend to 0 as ε tends to 0. Next, so me lengthy computations s how that P ε = − 1 2 P [  − ε  ε t 2 − 2 t  − ε  ε t + t 2  − ε  ε 1] + S P , Q ε = Q + P  − ε  ε 1 + R (  ε 1 +  − ε 1) − ˙ P [(  ε 1)(  ε t − t  ε 1)]+ ¨ P [(  ε 1)(  ε t 2 − 2 t  ε t + t 2  ε 1)] + ˙ R ( −  − ε t + t  − ε 1) + S Q , R ε = 1 2 P (  − ε  ε t − t  − ε  ε 1) + 1 2 R (  ε t −  − ε t + t  − ε 1 − t  ε 1) − 1 2 ˙ P [(  ε t − t  ε 1) 2 − (  ε 1)(  ε t 2 − 2 t  ε t + t 2  ε 1)] + S R . 12 S P , S Q and S R are three explicit matricial combinations of P, ˙ P , ¨ P , R , ˙ R, ¨ R which may b e r oughly upp er bounded as follows kS l k L ∞ ([ a,b ]) ≤ γ 2 ¯ N ( ε + χ I S ) 3 X ℓ =0 k P ( ℓ ) k L ∞ ([ a,b ]) + 3 X ℓ =0 k R ( ℓ ) k L ∞ ([ a,b ]) ! . Here, γ = max(1 , max ℓ | γ ℓ | ) and ¯ N = (2 N + 1)(4 N + 1) N 5 are tw o fixed num b ers w.r.t. ε . As a consequence, for all small δ > 0 , χ I S ( t ) and accor dingly S l ( t ) tend to 0 unifor mly in [ a + δ, b − δ ] as ε tends to 0. Finally , since ( c ) holds, insp e ction of each co efficient in the previous fo r mulas shows that P ε ( t ), Q ε ( t ), R ε ( t ) and ˜ Θ( x )( t ) tend resp ectively to P ( t ), Q ( t ) + ˙ R ( t ), R ( t ) − 1 2 ˙ P ( t ) and 0 as ε tends to 0 . The co nvergence o f D.E .L . is thus ensur ed for a ll la grang ia n L , for all function x and for all t ∈ [ a + δ , b − δ ], whic h ends the pr o of. R emark 6.1 . W e note that inside the safety interv al I S , for each op er ator  ε ∈ O N ,ε , the three formulas ho ld:  − ε  ε 1 = (  ε 1) 2 ,  − ε  ε t = t (  ε 1) 2 and  − ε  ε t 2 = t 2 (  ε 1) 2 + 2(  ε 1) N X ℓ = − N ℓ 2 c ℓ ! − 2(  ε t ) 2 . More generally , there exists po lynomial formulas for iterates of  ε acting o n the p olyno mials as expressions o f  ε t k for k ∈ N . R emark 6.2 . As men tioned in Rema rk 4.2, Cress o n defined extremal curves of the action as functions x ( t ) such that [ D A disc ( x )] ε = 0, that is to say the dominant par t of the F r´ echet deriv ative of the action is zero. Obviously , this do es not ensure that A disc is extremal a t x . The matter of the conv erg ence o f the Euler-La g range o p e rator restricted to extr emal curves of H¨ older ian regularity C β ([ a − ε , b + ε ] , R d ) for some β > 0 forms par t of the character iz a tion of Discrete Euler Lagrange equations. In con trast, we w ork here with a ll curv es x ∈ C 2 ([ a, b ] , R d ), not necessarily extremal. R emark 6 .3 . In the ter mino logy o f Γ-convergence, see for instance [7], the pre- ceding pro of implies that for all t 0 ∈ ] a, b [, if we denote ev t 0 the ev aluation map at t 0 , then the comp os ite mapping x → ev t 0 ◦ Θ( x ) is Γ-conv e r gent to x → ev t 0 ◦ ( − P ¨ x + ( − ˙ P + 2 R ) ˙ x + ( ˙ R + Q ) x − ˙ J 1 + J 2 ). Note that the authors in [7] prov ed the Γ-co nv er gence o f actions which ar e quadratic w.r.t. ˙ x and defined o n some spaces o f piecewise affine maps. References [1] J. Cresson , Non-differ en tiable va riational principles , J. Math. Anal. Appl., V ol. 30 7 (20 05), No. 1, pp. 4 8–64. [2] J. Cresson, G. F. F. Frederico and D . F. M. Torres , Constants of Motion for Non- D iffer entiable Quantum V ariational Pr oblems , T op o l. Metho ds No nlinear Anal., V ol. 33 (200 9 ), No . 2 , pp. 217–232 . 13 [3] V.M. Guibout and D.J. Scheeres , Solving two-p oint b oundary value pr oblems using gener ating funct ions: the ory and applic ations to astr o dy- namics , Gurfil, P . (ed.) Moder n Astro dynamics. Elsevier Astro dyna mics Series, Aca de mic Press (20 06) [4] K. Kime , Finite differ enc e appr oximation of c ontr ol via the p otential in a 1-D Schr o dinger e quation , Electr o n. J. Differen tial Equatio ns, V ol. 20 00 (2000), No. 2 6, pp. 1–10. [5] J. E. Marsden and M. West , Discr ete me chanics and variatio nal int e- gr ators , Acta Numer., V ol .10 (2001 ), pp. 357–514. [6] S. Ober-Bl ¨ obaum, O. Junge, J. E. Marsden , Discr ete Me chanics and Optimal Contr ol: an Analysis , ESAIM Cont ro l Optim. Calc. V a r., DOI: 10.105 1/co cv/ 2010 0 12 (2 0 10). [7] B. Schmidt, S. Leyendecker, M. Or tiz , Γ -c onver genc e of variational inte gr ators for c onstr aine d systems , J. Nonlinea r Sci., V ol. 19 (2009), No. 2, pp. 1 5 3–17 7. 14